{"schema_version":"ocme-site-results-v1","locale":"zh-Hant","page_type":"derivations","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"filters":{"q":"mko-convex-set"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-convex-set-01","title":{"zh-Hant":"凸集合：半空間與軸對齊盒的完整凸性證書，及有限點集的中點反例","en":"Convex set: complete convexity certificates for a halfspace and an axis-aligned box, and a midpoint counterexample for a finite point set"},"summary":{"zh-Hant":"在實向量空間 ℝ² 中：（一）證明閉半空間 H={(x,y) ∣ 2x−3y≤6} 是凸集合；（二）證明非空軸對齊盒 B=[−1,2]×[0,3] 是凸集合；兩個證明都必須對所有 p、q 與所有 λ∈[0,1] 成立。（三）用一個精確的中點見證否證有限點集 F={(0,0),(2,0),(0,2)} 的凸性，並說明為何抽查有限個 λ、或找不到反例，都不能證明凸性。","en":"In the real vector space ℝ²: (1) prove that the closed halfspace H={(x,y) ∣ 2x−3y≤6} is convex; (2) prove that the nonempty axis-aligned box B=[−1,2]×[0,3] is convex; both proofs must hold for all p, q and all λ∈[0,1]. (3) Refute the convexity of the finite point set F={(0,0),(2,0),(0,2)} with one exact midpoint witness, and explain why checking finitely many λ, or failing to find a counterexample, cannot prove convexity."},"mko_ids":["mko-convex-set"]}],"title":"概念推導導讀"}